I think this is a very common concern and certainly a worry of mine when I went into psychiatry. I clung onto this belief that I would be able to keep up with my medical knowledge, and I did the first two years, but once you move into the outpatient clinic you rapidly lose a lot of your medical knowledge. I used to be comfortable playing around the insulin, and I haven't done this is long enough that I have forgotten how to do this. It is just not possible to do it all and keep up with everything in the field. And given that in most systems we have access to specialists who can provide better care in their area of expertise, we should use this as needed. In the inpatient settings psychiatrists may manage basic medical problems like UTIs/STIs/cellulitis and so on, as well as hypertension, diabetes, hypercholesterolemia, hypothyroidism, asthma, COPD etc depending on the availability of other medical personnel. However if you are having to manage lots of medical problems then you will short-change your patients on the psychiatric aspects of their care though you will be overall responsible for their medical and psychiatric care. You will typically do physical examinations and lab tests as part of admissions and be responsible for management of any abnormalities in the first instance. I would rarely consult medicine when a resident in inpatient psychiatry.
In most outpatient settings it's not really appropriate to manage your patient's medical problems as they can easily see their primary care physician.* And it rarely makes sense to refill medications prescribed by another provider (they should be getting these refills from their providers). In particular systems such as the VA, I have refilled non-psych drugs for psych patients when they were hard to engage and it seemed reasonable enough. So yes, it can be appropriate. Also, if you order a UA on a pt and they have an uncomplicated UTI, I think it would be reasonable to prescribe a course of antibiotics. If your patients asks you for refills of their non-psych drugs it's at your discretion to decide whether it is appropriate or you feel comfortable to do so.
Psychiatrists are physicians first and psychiatrists second. Forget this at your peril. It just happens we focus on the psychological aspects of medicine and, if doing our jobs well, think more about the psychological, social, spiritual, ethical and legal dimensions of medical care. That does not mean we don't consider other aspects. We take more thorough histories than other specialists, and should perform a detailed mental status examination +/- cognitive state examination (which is part of the neurological examination), and come up with a thoughtful formulation of the problems the patients have and why they have those problems. In addition to treating psychiatric problems, it is often appropriate for us to treat medical complications of psychotropic drugs (e.g. tremor or acne from lithium, dystonia/dyskinesia with neuroleptics, erectile dysfunction with SSRIs, metabolic syndrome with atypical neuroleptics) and are in a good position to manage the behaviors that kill our patients like smoking, drinking alcohol, poor diet etc. I frequently perform thorough neurological examinations, order neuroimaging, and laboratory investigations on patients as part of their diagnostic work up. Because we have a better understanding of general medicine than other mental health professionals we have a better understanding of the psychological implications of physical illness, as well as the multifaceted ways medical problems can affect mental health, and when to formulate a more extensive medical differential for the psychiatric patient.
It can be hard to think about losing the skills acquired with being a physician that you have worked hard to earn, but as you go through psychiatric training you will hopefully just be developing those basic skills in a more specialized way and when you feel more confident with your abilities in treatment patients with more serious mental illness, and working psychotherapeutically with patients, you will worry less about having lost those skills which is inevitable regardless of what field of medicine you go into.
*an exception may be in PACT teams or homeless outreach teams for the chronically mentally ill who are hard to reach. In several of these that dont have primary care integrated, psychiatrists may be responsible for managing basic medical problems.